How do you find factor pairs using 4 5 6 8 that have a product between 3000 and 5000?

Answer 1

It is largely a trial and error exercise.

There are two sets of solutions: 1 digit x 3 digit factors or 2 digit x 2 digit factors.

Start with 1 digit x 3 digit

Suppose the 1 digit number is 4

then 3000 < 4*abc < 5000 where abc is a 3-digit number formed from {5,6,8}

Dividing each term of the above inequality by 4 implies that 750 < abc < 1250 so a must be greater than 7, that is a = 8. Both options for bc will work so the solutions are 4*856 = 3424 and 4*865 = 3460.

Continuing in a similar way, you will find 12 pairs of 1 digit x 3 digit factors.

You search in a similar way for the 2-digit by 2-digit except that now you also need to watch out for the factor pair in reverse. For example, 45*86 = 86*45 = 3870, but there is, in fact only one factor pair. You need to select factor pairs so that the first factor is smaller than the second. Of the 12 possible factor pairs, 8 will be within the required range.

That makes 20 pairs in all.

Answer 2

45 x 68 = 3060